Analysis of Parallel Algorithms for Matrix Chain Product and Matrix Powers on Distributed Memory Systems

Given N matrices A₁, A_2,...,A_N of size NtimesN, the matrix chain product problem is to compute A₁timesA₂times...timesA_N. Given an NtimesN matrix A, the matrix powers problem is to calculate the first N powers of A, that is, A, A², A³,..., A^N. We solve the two problems on distributed memory systems (DMSs) with p processors that can support one-to-one communications in T(p) time. Assume that the fastest sequential matrix multiplication algorithm has time complexity O(N^alpha), where the currently best value of a is less than 2.3755. Let p be arbitrarily chosen in the range 1lesplesN^alpha+1/(log N)². We show that the two problems can be solved by a DMS with p processors in T_chain(N,p)=O((N^alpha+1/p)+T(p))((N^2(2+1/alpha/p^2/alpha)(log+p/N)^1-2/alpha+log+((p log N)/N^alpha)) and T_power (N,p)=O(N^alpha+1/p+T(p)((N^2(1+1/alpha)/p^2/alpha)(log+p/2 log N)^1-2/alpha+(log N)²))) times, respectively, where the function log+ is defined as follows: log+ x=log x if xges1 and log+ x=1 if 0<x<1. We also give instantiations of the above results on several typical DMSs and show that computing matrix chain product and matrix powers are fully scalable on distributed memory parallel computers (DMPCs), highly scalable on DMSs with hypercubic networks, and not highly scalable on DMSs with mesh and torus networks.